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Number · Higher

Indices (fractional & negative)

Index laws extend to negative and fractional powers. A negative index means a reciprocal; a fractional index means a root. These appear frequently in Higher tier GCSE.

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Key facts to remember

  • 1a⁻ⁿ = 1/aⁿ (negative index = reciprocal)
  • 2a^(1/n) = ⁿ√a (unit fraction index = nth root)
  • 3a^(m/n) = (ⁿ√a)ᵐ — take the root first, then raise to the power.
  • 4a⁰ = 1 for any non-zero value of a.
  • 5Index laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ.
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Formulas

Negative index
a⁻ⁿ = 1 / aⁿ
Fractional index (unit)
a^(1/n) = ⁿ√a
Fractional index (general)
a^(m/n) = (ⁿ√a)ᵐ
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Worked examples

Example 1

Evaluate 8^(2/3)

Working

  1. The denominator 3 means cube root; the numerator 2 means square.
  2. ∛8 = 2
  3. 2² = 4
Answer4
Example 2

Evaluate 25^(−1/2)

Working

  1. Negative index: 25^(−1/2) = 1 / 25^(1/2)
  2. 25^(1/2) = √25 = 5
  3. 1 / 5 = 0.2
Answer1/5 (or 0.2)
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Common mistakes

✗Evaluating a^(m/n) as aᵐ ÷ n instead of (ⁿ√a)ᵐ.
✗Forgetting that a⁻ⁿ is a reciprocal — writing −aⁿ instead of 1/aⁿ.
✗Computing the power before the root when using a^(m/n) — always root first.
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Exam tips

✓For a^(m/n): the denominator is the root, the numerator is the power. Root first, then power — it keeps numbers small.
✓Write out each step: rewrite the negative or fractional index, evaluate the root, then apply the power.

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